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Improved bounds on \(m_r(2,q)q=19,25,27\). (English) Zbl 1295.51009

Summary: An \((n,r)\)-arc is a set of \(n\) points of a projective plane such that some \(r\), but no \(r+1\) of them, are collinear. The maximum size of an \((n,r)\)-arc in \(\mathrm{PG}(2, q)\) is denoted by \(m_r(2, q)\). In this paper, a new (286, 16)-arc in \(\mathrm{PG}(2,19)\), a new (341, 15)-arc, and a (388, 17)-arc in \(\mathrm{PG}(2,25)\) are constructed, as well as a (394, 16)-arc, a (501, 20)-arc, and a (532, 21)-arc in \(\mathrm{PG}(2,27)\). Tables with lower and upper bounds on \(m_r(2, 25)\) and \(m_r(2, 27)\) are presented as well. The results are obtained by nonexhaustive local computer search.

MSC:

51E20 Combinatorial structures in finite projective spaces
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